Is it a necessary non-epistemic truth? After all, it has a very lengthy partial proof in Principia Mathematica, and maybe they got something wrong. Perhaps you should check?
But then maybe you’re not using a formal system to prove it, but just taking it as an axiom or maybe as a definition of what “2” means using other symbols with pre-existing meanings. But then if I define the term “blerg” to mean “a breakfast product with non-obvious composition”, is that definition in itself a necessary truth?
Obviously if you mean “if you take one object and then take another object, you now have two objects” then that’s a contingent proposition that requires evidence. It probably depends upon what sorts of things you mean by “objects” too, so we can rule that one out.
Or maybe “necessary non-epistemic truth” means a proposition that you can “grok in fullness” and just directly see that it is true as a single mental operation? Though, isn’t that subjective and also epistemic? Don’t you have to check to be sure that it is one? Was it a necessary non-epistemic truth for you when you were young enough to have trouble with the concept of counting?
So in the end I’m not really sure exactly what you mean by a necessary truth that doesn’t need any checking. Maybe it’s not even a coherent concept.
Yes, such an agent will self-modify if it is presented with a Newcombe game before Omega determines how much money to put into boxes. It will even self-modify if there is a 1-in-1000 credence that Omega has not yet done so (or might change their mind).
At this point considerations come in such as what will happen if such an agent expects that they will face Newcombe-like games in the future but aren’t yet certain what form they will take or what the exact payoffs will be. Should they self-modify to something UDT-like now?